Alternately-optimized SNN method for acoustic scattering problem in unbounded domain
Haoming Song, Zhiqiang Sheng, Dong Wang, Junliang Lv
TL;DR
This work addresses acoustic scattering in unbounded domains by converting the exterior Helmholtz problem into a bounded-domain boundary-value problem using a transparent boundary condition based on the DtN operator. It introduces the alternately-optimized subspace neural network (AO-SNN), which trains a neural-network-based subspace with coefficients fixed at 1 to obtain robust basis functions, and then alternates with least-squares updates of the coefficients to achieve high-precision solutions. A novel mixed loss combines a metric term that enforces derivative-consistency with a standard PDE/boundary residual term, facilitating iterative refinement of the subspace and coefficients. Across numerical experiments, AO-SNN significantly outperforms PINN, RBDNN, and SNN, delivering relative $l^2$ errors down to the $10^{-7}$ level in bounded-domain tests and maintaining superior accuracy at higher wavenumbers, thereby providing a high-accuracy ML-based solver for unbounded-domain scattering problems, with future work aimed at further improving DtN truncation effects and scalability to large $\kappa$.
Abstract
In this paper, we propose a novel machine learning-based method to solve the acoustic scattering problem in unbounded domain. We first employ the Dirichlet-to-Neumann (DtN) operator to truncate the physically unbounded domain into a computable bounded domain. This transformation reduces the original scattering problem in the unbounded domain to a boundary value problem within the bounded domain. To solve this boundary value problem, we design a neural network with a subspace layer, where each neuron in this layer represents a basis function. Consequently, the approximate solution can be expressed by a linear combination of these basis functions. Furthermore, we introduce an innovative alternating optimization technique which alternately updates the basis functions and their linear combination coefficients respectively by training and least squares methods. In our method, we set the coefficients of basis functions to 1 and use a new loss function each time train the subspace. These innovations ensure that the subspace formed by these basis functions is truly optimized. We refer to this method as the alternately-optimized subspace method based on neural networks (AO-SNN). Extensive numerical experiments demonstrate that our new method can significantly reduce the relative $l^2$ error to $10^{-7}$ or lower, outperforming existing machine learning-based methods to the best of our knowledge.